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Empirical Efficiency Maximization:. Locally Efficient Covariate Adjustment in Randomized Experiments Daniel B. Rubin Joint work with Mark J. van der Laan. Outline. Review adjustment in experiments. Locally efficient estimation. Problems with standard methods.
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Empirical Efficiency Maximization: Locally Efficient Covariate Adjustment in Randomized Experiments Daniel B. Rubin Joint work with Mark J. van der Laan
Outline • Review adjustment in experiments. • Locally efficient estimation. Problems with standard methods. • New method addressing problems. • Abstract formulation. • Back to experiments, with simulations and numerical results. • Application to survival analysis.
Randomized Experiments • Example: Women recruited. Randomly assigned to diaphragm or no diaphragm. See if they get HIV. • Example: Men recruited. Randomly assigned to circumcision or no circumcision. See if they get HIV.
Randomized Experiments • Randomization allows causal inference. • No confounding. Differences in outcomes between treatment and control groups are due to the treatment. • Unverifiable assumptions needed for causal inference in observational studies.
Randomized Experiments with Covariates • Same setup, only now demographic or clinical measurements taken prior to randomization. • Question: With extra information, can we more precisely estimate causal parameters? • Answer: Yes. (Fisher, 1932). Subject’s covariate has information about how he or she would have responded in both arms.
Covariate Adjustment (has at least two meanings) 1: Gaining precision in randomized experiments. 2: Accounting for confounding in observational studies. This talk only deals with the first meaning.
Covariate Adjustment. • Not very difficult when covariates divide subjects into a handful of strata. • Have to think with a single continuous covariate (e.g. age). Modern studies can collect a lot of baseline information. Gene expression profile, complete medical history, biometric measurements. • Important longstanding problem, but lots of confusion. Not “solved.” Recent work by others.
Covariate Adjustment • Pocock et al. (2002) recently surveyed 50 clinical trial reports. • Of 50 reports, 36 used covariate adjustment for estimating causal parameters, and 12 emphasized adjusted over unadjusted analysis. • “Nevertheless, the statistical emphasis on covariate adjustment is quite complex and often poorly understood, and there remains confusion as to what is an appropriate statistical strategy.”
Recent Work on this Problem • Koch et al. (1998). - Modifications to ANCOVA. • Tsiatis et al. (2000, 2007a, 2007b). - Locally efficient estimation. • Moore and van der Laan (2007). - Targeted maximum likelihood. • Freedman (2007a, 2007b). - Classical methods under misspecification.
Covariate Adjustment • Can choose to ignore baseline measurements. Why might extra precision be worth it? • Narrow confidence intervals for treatment effect. • Stop trials earlier. • Subgroup analysis, having smaller sample sizes.
Locally Efficient Estimation • Primarily motivated by causal inference problems in observational studies. • Origin in Robins and Rotnitzky (1992), Robins, Rotnitzky, and Zhao (1994). • Surveyed in van der Laan and Robins (2003), Tsiatis (2006).
Locally Efficient Estimation in Randomized Experiments • Working model for treatment distribution given covariates known by design. • So what does local efficiency tell us? • Model outcome distribution given (covariates, treatment). • We’ll be asymptotically efficient if working model is correct, but still asymptotically normal otherwise. • But what does this mean if there’s no reason to believe the working model? Unadjusted estimators are also asymptotically normal. What about precision?
Empirical Efficiency Maximization • Working model for outcome distribution given (treatment, covariates) typically fit with likelihood-based methods. • Often linear, logistic, or Cox regression models. • “Factorization of likelihood” means such estimates lead to double robustness in observational studies. But such robustness is superfluous in controlled experiments. • We try to find the working model element resulting in the parameter estimate with smallest asymptotic variance.
Connection to High Dimensional or Complex Data • Suppose a high dimensional covariate is related to the outcome, and we would like to adjust for it to gain precision. • Many steps in data processing can be somewhat arbitrary (e.g. dimension reduction, smoothing, noise removal). • With cross-validation, new loss function can guide selection of tuning parameters governing this data processing.
1=Unadjusted estimator ignoring covariates. 2=Likelihood-based locally efficient. 3=Empirical efficiency maximization. 4=Efficient.
1=Unadjusted difference in means. 2=Standard likelihood-based locally efficient estimator. 3=empirical efficiency maximization. 4=efficient estimator.
Sneak Peak: Survival Analysis • Form locally efficient estimate. Working model for full data distribution now likely a proportional hazards model. • For estimating (e.g. five-year survival), will be asymptotically efficient if model correct. Otherwise still asymptotically normal. • But locally efficient estimator can be worse than Kaplan-Meier if model is wrong.
Sneak Peak: Survival Analysis. • Generated Data. Want five-year survival.
1=Kaplan-Meier. 2=Likelihood-based locally efficient. 3=Empirical efficiency maximization. 4=Efficient.
Summary • Robins and Rotnitzky’s locally efficient estimation developed for causal inference in observational studies. • In experiments, estimators can gain or lose efficiency, depending on validity of working model. • Often there might be no reason to have any credence in a working model. • A robustness result implies we can better fit working model (or select tuning parameters in data processing) with a nonstandard loss function (the squared influence curve).
